How to get precision when rotating / positioning textures?
Edit1:
For conveniance I build a Group-Node that does the centering, and also allows to plug in Rot_Vectors.
This was done by combining the help from:
    Cycles mapping node - rotation pivot
    How to add a socket for mapping node?
    Can I make a custom node from a group and have that available in other projects?  

This is how the node is used:


This is the inside of the node, with the Translate and Rotate node from gandalf3:


And here an optional Driver Setup to controll your rotation with an gameobject in the scene. Maby you could build yourselfe a GameObject that Visualizes your Rotation in the Viewport if you want to go further.


Edit_0:
If there are no other reasons you dont want to use a mapping node i would recomend the material view for visual comfort.  

The reason you dont get any rotation value in a panel as you get in Object-Mode is that there are no objects, just verts in the u-v-coordinate system those are represented as vector2 -> u and v. Its the same in Edit-Mode, there the vers are also just points as Vector3. Rotation can be done as an operation but to store those blender oughta be node-based and also produce temporary objects of specifiec groups of verts that had been rotated.  

I would recomend you to unwrap you upper and lower face together so they are exacly ontop each other and "rotated" the same and use a mapping node in Material View.  

maby an option: orthogonal upsidedown camera to your coin and "Project from View" as unrwaping so your uv is axis-aligned